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Descent Directions and Efficient Solutions in Discretely Distributed Stochastic Programs

  • Book
  • © 1988

Overview

Authors:
  1. Kurt Marti

    1. Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr München, Neubiberg, Germany

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Part of the book series: Lecture Notes in Economics and Mathematical Systems (LNE, volume 299)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XIV
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  2. Stochastic programs with a discrete distribution

    • Kurt Marti
    Pages 1-3

  3. Stochastic dominance (SD) and the construction of feasible descent directions

    • Kurt Marti
    Pages 4-17

  4. Convex programs for solving (3.1)–(3.4a),(3.5)

    • Kurt Marti
    Pages 18-23

  5. Stationary points (efficient solutions) of (SOP)

    • Kurt Marti
    Pages 24-30

  6. Optimal solutions of (P X,D), \(({\tilde P_{x,D}})\)

    • Kurt Marti
    Pages 31-38

  7. Optimal solutions (y*,T*) of \(\left( {\tilde P_{X,D}^Q} \right)\) having τij*>0 for all i∈S,j∈R

    • Kurt Marti
    Pages 39-42

  8. Existence of solutions of the SD-conditions (3.1)–(3.5), (12.1)–(12.5), resp.; Representation of stationary points

    • Kurt Marti
    Pages 43-85

  9. Construction of solutions (y,T) of (12.1)–(12.4) by means of formula (44)

    • Kurt Marti
    Pages 86-132

  10. Construction of solutions (y,B) of (46) by using representation (60) of (A(ω),b(ω))

    • Kurt Marti
    Pages 133-168

  11. Back Matter

    Pages 169-183
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Keywords

About this book

In engineering and economics a certain vector of inputs or decisions must often be chosen, subject to some constraints, such that the expected costs arising from the deviation between the output of a stochastic linear system and a desired stochastic target vector are minimal. In many cases the loss function u is convex and the occuring random variables have, at least approximately, a joint discrete distribution. Concrete problems of this type are stochastic linear programs with recourse, portfolio optimization problems, error minimization and optimal design problems. In solving stochastic optimization problems of this type by standard optimization software, the main difficulty is that the objective function F and its derivatives are defined by multiple integrals. Hence, one wants to omit, as much as possible, the time-consuming computation of derivatives of F. Using the special structure of the problem, the mathematical foundations and several concrete methods for the computation of feasible descent directions, in a certain part of the feasible domain, are presented first, without any derivatives of the objective function F. It can also be used to support other methods for solving discretely distributed stochastic programs, especially large scale linear programming and stochastic approximation methods.

Authors and Affiliations

  • Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr München, Neubiberg, Germany

    Kurt Marti

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